In this paper, a novel hybrid semi-implicit meshless weighted least-squares (WLS) scheme H-SIFPM is developed for the first time by coupling the semi-implicit finite point-set method and finite difference method (FDM), and then applied to predict the time-memory nonlinear behavior dominated by a 2D variable-order time-fractional nonlinear Schrödinger equation (TF-NLSE). The proposed H-SIFPM for TF-NLSE is mainly derived from that: firstly, the variable-order time-fractional with Caputo derivative is approximated by a FDM scheme and then imposed to the approximation of WLS as a boundary condition; secondly, the hybrid process is implicitly discretized in temporal level, and an implicit meshless FPM scheme is developed; thirdly, an iterative technique is adopted to treat the implicit format. Subsequently, the estimated error and stability of the proposed scheme is tentatively proved. Moreover, the accuracy and capacity of the proposed H-SIFPM are also tested by solving several examples with constant/variable-order cases, in which a second-order numerical convergent rate is illustrated. We also demonstrate the advantages of the meshless method by solving the problem on complex irregular domain with local refinement. Finally, the nonlinear dispersion behavior dominated by 2D TF-NLSE/TF-GPE with different boundary conditions are predicted, and the FDM results are also presented for comparison. All numerical results show that the proposed H-SIFPM scheme for TF-NLSEs is stable, accurate and flexible.

中文翻译：

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一种新颖的半隐式 WLS 方案，适用于 2D 变阶 TF-NLSE 中的时间记忆非线性行为


本文通过耦合半隐式有限点集方法和有限差分法（FDM），首次提出了一种新型混合半隐式无网格加权最小二乘（WLS）方案H-SIFPM，并应用于预测由二维变阶时间分数非线性薛定谔方程 (TF-NLSE) 主导的时间记忆非线性行为。所提出的用于TF-NLSE的H-SIFPM主要源于：首先，通过FDM方案来近似具有Caputo导数的变阶时间分数，然后将其作为边界条件施加到WLS的近似中；其次，在时间层面对混合过程进行隐式离散化，并提出隐式无网格FPM方案；第三，采用迭代技术来处理隐式格式。随后，初步证明了该方案的估计误差和稳定性。此外，还通过求解几个具有常/变阶情况的例子来测试所提出的H-SIFPM的精度和容量，其中说明了二阶数值收敛率。我们还通过局部细化解决复杂不规则域上的问题来展示无网格方法的优点。最后，预测了不同边界条件下2D TF-NLSE/TF-GPE主导的非线性色散行为，并给出了FDM结果进行比较。所有数值结果表明，所提出的 TF-NLSE 的 H-SIFPM 方案稳定、准确且灵活。